The Lyapunov and Lasalle Theorems

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8 Οκτ 2013 (πριν από 4 χρόνια και 9 μήνες)

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The Lyapunov and Lasalle Theorems
Hypothesis Suppose that X

is an equilibrium of the system X
= F(x) with F contin-
uously differentiable.
Definition 1 A continuously differentiable real valued function L from on an open neigh-
borhood O 3 X

is a lyapunov function when it has the following two properties.
i.L > 0 on On X

and L(X

) = 0.
ii r
L(X)  F(X)  0 on O.
It is a strict lyapunov function when in addition
L(X)  F(X) < 0 on On X

Property ii asserts that L is nondecreasing on orbits in O.Property iii asserts that the
time derivative of L on orbits in On X

is strictly negative.
The first two theorems are due to Lyapunov.The last two are called LaSalle’s Invariance
Theorem 1 If there exists a lyapunov function,then the equilibrium X

is stable
Theorem 2 If there exists a strict lyapunov function,then the equilibrium X

is asymp-
totically stable
Theorem 3 Supposse that L is a Lyapunov functional on O and X(t) is an orbit lying
in a closed bounded set K  O.If Z
is an!-limit point of X(t) and Z(t) is the orbit
with Z(0) = Z
,then Z(t) lies in K and L(Z(t)) is independent of t for t  0.
Theorem 4 Suppose that L is a lyapunov functional on O and that P  O is a closed
bounded set satisfying
i.For each t  0,
(P)  P where 
is the flow of the differential equation.

is the only orbit in P along which L is constant for t  0.
Then every orbit starting in P converges to X

as t!1.
Equivalently,the basin of attraction of X

contains P.In many examples the set P is of
the form fL  g.
Theorem 4 and Theorem 2 follow quickly from Theorem 3